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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
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\language american
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\end_header
\begin_body
\begin_layout Title
Boltzmann Generators
\end_layout
\begin_layout Section
Introduction
\end_layout
\begin_layout Standard
Equilibrium statistical mechanics is concerned with computing the statistical
properties of an ensemble, i.e.
infinitely many copies, of a microscopic physical system.
A classical example is the Ising magnetization model, where interesting
quantities are which fraction of spins are
\begin_inset Quotes eld
\end_inset
up
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
down
\begin_inset Quotes erd
\end_inset
for a given external field, or spatial properties, such as the typical
size of contiguous clusters of equal spins (Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:problem_description"
plural "false"
caps "false"
noprefix "false"
\end_inset
a).
A second example is protein biophysics – for a protein system that can
exist in two or more macroscopic states (active or inactive, folded or
unfolded, bound or dissociated), what is the probability of finding the
protein in either of these states, and does their population depend on
external factors, such as temperature, illumination or electric fields
(Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:problem_description"
plural "false"
caps "false"
noprefix "false"
\end_inset
b).
\end_layout
\begin_layout Standard
A common concept to approach these problems is to assign to each possible
configuration
\begin_inset Formula $\mathbf{x}$
\end_inset
(the setting of all spins, the position of all protein atoms, etc.) a dimensionl
ess energy,
\begin_inset Formula $u(\mathbf{x})$
\end_inset
whose contributions depend on the thermodynamic ensemble of interest (e.g.,
constant particle number, constant volume, etc.
– see SI).
Then, each configuration has the equilibrium probability:
\begin_inset Formula
\[
\mu(\mathbf{x})=\frac{1}{Z}\mathrm{e}^{-u(\mathbf{x})}.
\]
\end_inset
Now we would like to compute expectation values of relevant observables
weighted by this probability distribution, such as the probability of finding
the Ising spins
\begin_inset Quotes eld
\end_inset
up
\begin_inset Quotes erd
\end_inset
or the protein in the active state.
However, following this idea is fraught with difficulties that inspire
much of the research done in statistical mechanics.
Even if
\begin_inset Formula $u(\mathbf{x})$
\end_inset
is exactly known, it if often very expensive to evaluate as it contains
all microscopic interactions between spins or atoms, possible involving
millions of terms.
The normalization factor,
\begin_inset Formula $Z$
\end_inset
, is an integral of
\begin_inset Formula $\mathrm{e}^{-u(\mathbf{x})}$
\end_inset
of all possible configurations
\begin_inset Formula $\mathbf{x}$
\end_inset
, and generally considered to be impossible to compute for large systems
with nontrivial interactions.
\end_layout
\begin_layout Standard
The only known strategies to tackle this problem are Markov-Chain Monte
Carlo (MCMC) simulations where we propose changes to
\begin_inset Formula $\mathbf{x}$
\end_inset
(e.g., flipping a spin) and accepting or rejecting according to how the energy
changes, or Molecular Dynamics (MD) simulations where we change
\begin_inset Formula $\mathbf{x}$
\end_inset
by a tiny step that involves the derivatives of the energy with respect
to
\begin_inset Formula $\mathbf{x}$
\end_inset
that ensure that
\begin_inset Formula $\mu(\mathbf{x})$
\end_inset
will be sampled.
These methods are generally extremely expensive and much of the worldwide
supercomputing resources are used for MCMC or MD simulations.
This expense is due to (1) evaluating
\begin_inset Formula $u(\mathbf{x})$
\end_inset
or its gradient which may involve computing millions of interaction terms
that make every step expensive, and (2) computing expectation values according
to
\begin_inset Formula $\mu(\mathbf{x})$
\end_inset
involves sampling back and forth between phases or states that need and
extremely large number of steps (e.g.
\begin_inset Formula $10^{9}-10^{15}$
\end_inset
steps in a typical MD simulation to fold or unfold a protein).
Only for some systems we know specifically designed MCMC moves which make
large changes in an efficient way (e.g.
cluster moves in Ising models or implicit protein models [
\series bold
cite
\series default
]).
Speeding up the transition with enhanced sampling methods is possible if
we can
\begin_inset Quotes eld
\end_inset
drive
\begin_inset Quotes erd
\end_inset
the system along a few collective coordinates which must describe all the
slow transitions of the system and sampling the remaining fast motions
[
\series bold
cite metadynamics etc
\series default
], but this approach breaks down in systems where the number of slow processes
is large.
\end_layout
\begin_layout Standard
Ideally, we would like to have a machine that samples
\begin_inset Formula $\mathbf{x}$
\end_inset
directly from the distribution
\begin_inset Formula $\mu(\mathbf{x})$
\end_inset
, or at least something very close to it.
This problem is probably impossible to solve in configuration space, because
for a large dimension, the subvolume of low-energy configurations is vanishingl
y small compared to the full configuration space and has a complex and unknown
shape.
Thus, generating
\begin_inset Formula $\mathbf{x}$
\end_inset
by simply generating random configurations is bound to fail – e.g.
generating random atom positions in a box will have almost zero probability
to generate a configuration that corresponds to a protein (Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:problem_description"
plural "false"
caps "false"
noprefix "false"
\end_inset
b), and almost always result in numerically infinite energies that contribute
nothing to
\begin_inset Formula $\mu(\mathbf{x})$
\end_inset
.
\end_layout
\begin_layout Standard
Nonetheless, we directly address this problem here.
Our strategy is: since sampling
\begin_inset Formula $\mu(\mathbf{x})$
\end_inset
in configuration space is too difficult, can we instead find a coordinate
transformation of
\begin_inset Formula $\mathbf{x}$
\end_inset
to another representation
\begin_inset Formula $\mathbf{z}$
\end_inset
, in which sampling is easy and every sample can be back-transformed to
a relevant configuration
\begin_inset Formula $\mathbf{x}$
\end_inset
that contributes to
\begin_inset Formula $\mu(\mathbf{x})$
\end_inset
?
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
a)
\begin_inset Graphics
filename figs/intro_ising.jpg
lyxscale 50
width 30text%
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
hspace{0.2cm}
\end_layout
\end_inset
b)
\begin_inset Graphics
filename figs/intro_confchange.jpg
lyxscale 50
width 50text%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:problem_description"
\end_inset
State- and phase transitions in complex metastable systems
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Section
Invertible Networks
\end_layout
\begin_layout Standard
To find such a transformation, we employ machine learning, specifically
deep learning that has recently led to breakthroughs in pattern recognition,
games and autonomous control [
\series bold
cite
\series default
].
The key idea is to sample a complicated distribution
\begin_inset Formula $p_{X}(\mathbf{x})\propto\mathrm{e}^{-u(\mathbf{x})}$
\end_inset
by learning a reversible transformation to a latent space,
\begin_inset Formula $\mathbf{z}=T_{xz}(\mathbf{x})$
\end_inset
, such that
\begin_inset Formula $p_{Z}(\mathbf{z})=p_{Z}\left(T_{xz}(\mathbf{x})\right)$
\end_inset
is simple.
Specifically, we want to make the distribution in
\begin_inset Formula $z$
\end_inset
a standard normal distribution
\begin_inset Formula $p_{Z}(\mathbf{z})=\mathcal{N}\left(0,\mathbf{I}\right)$
\end_inset
.
\end_layout
\begin_layout Standard
We call the transformation from configuration to latent space
\begin_inset Formula $T_{xz}$
\end_inset
and the inverse transformation
\begin_inset Formula $T_{zx}=T_{xz}^{-1}$
\end_inset
.
In general, these transformations are not volume preserving and we thus
keep record of the Jacobian of the transformation.
We use the notation:
\begin_inset Formula
\begin{align*}
\mathbf{J}_{zx} & =\frac{\partial T_{zx}(\mathbf{z})}{\partial\mathbf{z}^{\top}}\\
\mathbf{J}_{xz} & =\frac{\partial T_{xz}(\mathbf{x})}{\partial\mathbf{x}^{\top}}
\end{align*}
\end_inset
Random variables are transformed according to:
\begin_inset Formula
\begin{align}
p_{X}(\mathbf{x}) & =p_{Z}(\mathbf{z})\left|\mathbf{J}_{zx}(\mathbf{z})\right|^{-1}\label{eq:transform_zx}\\
p_{Z}(\mathbf{z}) & =p_{X}(\mathbf{x})\left|\mathbf{J}_{xz}(\mathbf{x})\right|^{-1}\label{eq:transform_xz}
\end{align}
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename figs/network_structure2.pdf
width 100text%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig_network-architecture"
\end_inset
Network architecture
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
NICE and NICER
\end_layout
\begin_layout Standard
We first use the volume-preserving transformation proposed for nonlinear
independent components estimation (NICE, Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "fig_network-architecture"
plural "false"
caps "false"
noprefix "false"
\end_inset
)
\begin_inset CommandInset citation
LatexCommand cite
key "DinhDruegerBengio_NICE2015"
literal "false"
\end_inset
.
For this transformation one defines two groups of variables,
\begin_inset Formula $\mathbf{x}_{1}$
\end_inset
and
\begin_inset Formula $\mathbf{x}_{2}$
\end_inset
, and employs a nonlinear transformation,
\begin_inset Formula $P$
\end_inset
, to transform only
\begin_inset Formula $\mathbf{x}_{2}$
\end_inset
, while
\begin_inset Formula $\mathbf{x}_{1}$
\end_inset
is unchanged.
Independent of the choice of
\begin_inset Formula $P$
\end_inset
, this transformation is easily invertible:
\begin_inset Formula
\[
\begin{array}{ccc}
\begin{aligned}\mathbf{y}_{1} & =\mathbf{x}_{1}\\
\mathbf{y}_{2} & =\mathbf{x}_{2}+P(\mathbf{x}_{1})
\end{aligned}
& & \begin{aligned}\mathbf{x}_{1} & =\mathbf{y}_{1}\\
\mathbf{x}_{2} & =\mathbf{y}_{2}-P(\mathbf{y}_{1})
\end{aligned}
\end{array}
\]
\end_inset
This transformation has the following Jacobian:
\begin_inset Formula
\[
\mathbf{J}_{xy}=\left(\begin{array}{cc}
1 & 0\\
\frac{\partial P(x_{1})}{\partial x_{1}} & 1
\end{array}\right),
\]
\end_inset
and, as a result:
\begin_inset Formula
\begin{align*}
\left|\det\left(\mathbf{J}_{xy}\right)\right| & =1\\
\left|\det\left(\mathbf{J}_{yx}\right)\right| & =1
\end{align*}
\end_inset
\end_layout
\begin_layout Standard
This makes the transformation volume-preserving.
In physics, such transformations are found in symplectic integrators, and
in incompressible fluid flows.
\end_layout
\begin_layout Standard
In order to transform both channels, we define the NICER layer (Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "fig_network-architecture"
plural "false"
caps "false"
noprefix "false"
\end_inset
) with involves a second transformation
\begin_inset Formula $Q$
\end_inset
:
\begin_inset Formula
\[
\begin{array}{ccc}
\begin{aligned}\mathbf{z}_{2} & =\mathbf{y}_{2}\\
\mathbf{z}_{1} & =\mathbf{y}_{1}+Q(\mathbf{y}_{2})
\end{aligned}
& & \begin{aligned}\mathbf{y}_{2} & =\mathbf{z}_{2}\\
\mathbf{y}_{1} & =\mathbf{z}_{1}-Q(\mathbf{z}_{2})
\end{aligned}
\end{array}
\]
\end_inset
By concatenating many of these layers, we obtain the deep NICER network
\begin_inset Formula $T_{xz}$
\end_inset
is reversible
\begin_inset Formula $T_{zx}=T_{xz}^{-1}$
\end_inset
and volume-preserving as well (Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "fig_network-architecture"
plural "false"
caps "false"
noprefix "false"
\end_inset
).
Here we use two-layer perceptrons with 100 hidden neurons and rectified
linear units [
\series bold
cite
\series default
] for each
\begin_inset Formula $P$
\end_inset
and
\begin_inset Formula $Q$
\end_inset
and ten NICER layers.
\end_layout
\begin_layout Standard
As a result of volume preservation, the transformation of probability densities
is trivial:
\begin_inset Formula
\begin{align*}
\log p_{X}(x) & =\log p_{Z}(T_{xz}(\mathbf{x}))\\
\log p_{Z}(z) & =\log p_{X}(T_{zx}(\mathbf{z}))
\end{align*}
\end_inset
\end_layout
\begin_layout Subsection
Scaling layer
\end_layout
\begin_layout Standard
We generalize the transformation by adding a scaling layer:
\begin_inset Formula
\begin{align*}
\mathbf{z} & =T_{xz}(\mathbf{x})=\mathbf{s}\circ\mathbf{x}\\
\mathbf{x} & =T_{zx}(\mathbf{z})=\mathbf{s}^{-1}\circ\mathbf{z}
\end{align*}
\end_inset
where
\begin_inset Formula $\mathbf{s}=(s_{1},...,s_{n})$
\end_inset
are the scaling factors and
\begin_inset Formula $\mathbf{s}^{-1}=(s_{1}^{-1},...,s_{n}^{-1})^{T}$
\end_inset
.
The Jacobians of this transformation are:
\begin_inset Formula
\begin{align*}
\left|\det\left(\mathbf{J}_{xz}\right)\right| & =\left|\det\left(\mathrm{diag}(\mathbf{s})\right)\right|=\left|\prod_{i}s_{i}\right|\\
\left|\det\left(\mathbf{J}_{zx}\right)\right| & =\left|\det\left(\mathrm{diag}(\mathbf{s}^{-1})\right)\right|=\left|\prod_{i}s_{i}^{-1}\right|
\end{align*}
\end_inset
Unless a better initialization for the problem at hand is available, we
recomment to initialize the network in a regime with low condition number
by choosing
\begin_inset Formula
\[
\mathbf{s}^{(0)}=\mathrm{diag}(1,...,1).
\]
\end_inset
\end_layout
\begin_layout Standard
The scaling layer transforms logarithmized probability distributions as
follows:
\begin_inset Formula
\begin{align*}
\log p_{X}(x) & =\left|\sum_{i}\log s_{i}\right|+\log p_{Z}(\mathbf{s}\circ\mathbf{x})\\
\log p_{Z}(z) & =\left|-\sum_{i}\log s_{i}\right|+\log p_{X}(\mathbf{s}^{-1}\circ\mathbf{z})
\end{align*}
\end_inset
\end_layout
\begin_layout Subsection
Exponential Scaling Layer
\end_layout
\begin_layout Standard
When scaling is just used to stretch or compress space and it is not desired
to change signs, we can choose the following parametrization of
\begin_inset Formula $\mathbf{S}$
\end_inset
which enforces nonnegativity of the scaling factors:
\begin_inset Formula
\[
\mathbf{S}=\mathrm{diag}\left(\exp(k_{1}),...,\exp(k_{1})\right),
\]
\end_inset
where
\begin_inset Formula $k_{i}$
\end_inset
are the trainable parameters.
With this formulation, the Jacobians become:
\begin_inset Formula
\begin{align*}
\left|\det\left(\mathbf{J}_{xz}\right)\right| & =\exp\left(\sum_{i}k_{i}\right)\\
\left|\det\left(\mathbf{J}_{zx}\right)\right| & =\exp\left(-\sum_{i}k_{i}\right)
\end{align*}
\end_inset
Note that the absolute value operator is no longer needed as the value of
the exponential function is always nonnegative.
The exponential scaling layer transforms logarithmized probability distribution
s as:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align*}
\log p_{X}(x) & =\sum_{i}k_{i}+\log p_{Z}(\exp(\mathbf{k})\circ\mathbf{x})\\
\log p_{Z}(z) & =-\sum_{i}k_{i}+\log p_{X}(\exp(-\mathbf{k})\circ\mathbf{z})
\end{align*}
\end_inset
\lang english
\begin_inset Note Note
status open
\begin_layout Plain Layout
In general:
\begin_inset Formula
\begin{align*}
p_{X}(x) & =\left|\det\left(\frac{dT_{xz}}{dx}\right)\right|p_{Z}(T_{xz}(x))=\left|\det\left(\frac{dT_{zx}}{dx}\right)\right|^{-1}p_{Z}(T_{xz}(x))\\
p_{Z}(z) & =\left|\det\left(\frac{dT_{zx}}{dz}\right)\right|p_{X}(T_{zx}(z))=\left|\det\left(\frac{dT_{xz}}{dz}\right)\right|^{-1}p_{X}(T_{zx}(z))
\end{align*}
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\lang english
\begin_inset Formula
\begin{align*}
p_{X}(\mathbf{x}) & =\left|\det\left(S\right)\right|^{-1}p_{Z}(T_{xz}(\mathbf{x}))
\end{align*}
\end_inset
\end_layout
\begin_layout Standard
\lang english
If we simply decide to sample
\begin_inset Formula $z$
\end_inset
from a normal distribution and transform to
\begin_inset Formula $x$
\end_inset
, the network represents the density
\begin_inset Formula
\begin{align}
p_{X}(\mathbf{x}) & =\left|\det\left(S\right)\right|^{-1}\mathcal{N}(0,\mathbf{I}).\label{eq:pX_withS_from_Normal}
\end{align}
\end_inset
\end_layout
\begin_layout Subsection
RealNVP
\end_layout
\begin_layout Standard
Forward transformation:
\begin_inset Formula $\mathbf{x}\rightarrow\mathbf{y}\rightarrow\mathbf{z}$
\end_inset
with the first step:
\begin_inset Formula
\begin{align*}
\mathbf{y}_{1} & =\mathbf{x}_{1}\\
\mathbf{y}_{2} & =\mathbf{x}_{2}\odot\exp\left(S(\mathbf{x}_{1})\right)+T(\mathbf{x}_{1})
\end{align*}
\end_inset
and the Jacobian:
\begin_inset Formula
\begin{align*}
\mathbf{J}_{xy} & =\left[\begin{array}{cc}
I & 0\\
\frac{\partial\mathbf{y}_{2}}{\partial\mathbf{x}_{1}} & \mathrm{diag}\left[\exp\left(S(\mathbf{x}_{1})\right)\right]
\end{array}\right]\\
\left|\det\left(\mathbf{J}_{xy}\right)\right| & =\mathrm{e}^{\sum_{i}S_{i}(\mathbf{x}_{1})}
\end{align*}
\end_inset
Likelihood:
\begin_inset Formula
\[
p_{X}(\mathbf{x})=J(\mathbf{x})p_{Z}(T_{xz}(\mathbf{x}))
\]
\end_inset
with
\begin_inset Formula
\[
J(\mathbf{x})=\left|\det\left(\frac{dT_{xz}}{dx}\right)\right|
\]
\end_inset
\end_layout
\begin_layout Standard
The Log Likelihood of a Gaussian in
\begin_inset Formula $\mathbf{z}$
\end_inset
with mean 0 and std
\begin_inset Formula $\sigma$
\end_inset
is given by:
\begin_inset Formula
\begin{align*}
L(\mathbf{x}) & =\log J(\mathbf{x})-n\log\left(\sigma\right)-\frac{1}{2\sigma^{2}}\mathbf{z}^{\top}\mathbf{z}+\mathrm{const}\\
& =\log J(\mathbf{x})-\frac{1}{2\sigma^{2}}\mathbf{z}^{\top}\mathbf{z}+\mathrm{const}
\end{align*}
\end_inset
For multiple trajectories that should each be standartized, if we sum their
log-likelihoods we will get the same loss function except for a constant
pre-factor.
If we instead sum their likelihoods, we get:
\begin_inset Formula
\begin{align*}
\mathrm{e}^{L(\mathbf{X})} & =\mathrm{e}^{\sum_{s=1}^{N_{1}}\log J(\mathbf{x}_{s})-\frac{1}{2\sigma^{2}}\mathbf{z}_{s}^{\top}\mathbf{z}_{s}}+\mathrm{e}^{\sum_{s=1}^{N_{2}}\log J(\mathbf{x}_{t})-\frac{1}{2\sigma^{2}}\mathbf{z}_{t}^{\top}\mathbf{z}_{t}}\\
L(\mathbf{X}) & =\log\left(\mathrm{e}^{\sum_{s=1}^{N_{1}}\log J(\mathbf{x}_{s})-\frac{1}{2\sigma^{2}}\mathbf{z}_{s}^{\top}\mathbf{z}_{s}}+\mathrm{e}^{\sum_{s=1}^{N_{2}}\log J(\mathbf{x}_{t})-\frac{1}{2\sigma^{2}}\mathbf{z}_{t}^{\top}\mathbf{z}_{t}}\right)\\
& =\mathrm{logsumexp}\left(\sum_{s=1}^{N_{1}}\log J(\mathbf{x}_{s})-\frac{1}{2\sigma^{2}}\mathbf{z}_{s}^{\top}\mathbf{z}_{s},\sum_{s=1}^{N_{2}}\log J(\mathbf{x}_{s})-\frac{1}{2\sigma^{2}}\mathbf{z}_{s}^{\top}\mathbf{z}_{s}\right)
\end{align*}
\end_inset
\begin_inset Formula
\begin{align*}
L(\mathbf{X}) & =\sum_{s=1}^{N_{1}}\left(\frac{1}{N_{1}}\log J(\mathbf{x}_{s})-\frac{1}{N_{1}}\frac{1}{2\sigma^{2}}\mathbf{z}_{s}^{\top}\mathbf{z}_{s}\right)+\sum_{t=1}^{N_{2}}\left(\frac{1}{N_{2}}\log J(\mathbf{x}_{t})-\frac{1}{N_{2}}\frac{1}{2\sigma^{2}}\mathbf{z}_{t}^{\top}\mathbf{z}_{t}\right)
\end{align*}
\end_inset
If we have equally many trajectories (
\begin_inset Formula $N/2$
\end_inset
) in each batch, we can concatenate them and write
\begin_inset Formula
\begin{align*}
L(\mathbf{X}) & =\sum_{t=1}^{N/2}\left(\frac{2}{N}\log J(\mathbf{x}_{s})-\frac{2}{N}\frac{1}{2\sigma^{2}}\mathbf{z}_{s}^{\top}\mathbf{z}_{s}\right)+\sum_{t=N/2}^{N}\left(\frac{2}{N}\log J(\mathbf{x}_{t})-\frac{2}{N}\frac{1}{2\sigma^{2}}\mathbf{z}_{t}^{\top}\mathbf{z}_{t}\right)\\
& =\frac{2}{N}\sum_{t=1}^{N}\log J(\mathbf{x}_{s})-\frac{2}{N}\sum_{t=1}^{N}\frac{1}{2\sigma^{2}}\mathbf{z}_{s}^{\top}\mathbf{z}_{s}
\end{align*}
\end_inset
\end_layout
\begin_layout Standard
Reverse transformation:
\begin_inset Formula
\begin{align*}
\mathbf{x}_{1} & =\mathbf{y}_{1}\\
\mathbf{x}_{2} & =\left(\mathbf{y}_{2}-T(\mathbf{x}_{1})\right)\odot\exp\left(-S(\mathbf{y}_{1})\right)
\end{align*}
\end_inset
and the Jacobian:
\begin_inset Formula
\begin{align*}
\frac{\partial\mathbf{x}}{\partial\mathbf{y}} & =\left[\begin{array}{cc}
I & 0\\
\frac{\partial\mathbf{x}_{2}}{\partial\mathbf{y}_{1}} & \mathrm{diag}\left[\exp\left(-S(\mathbf{y}_{1})\right)\right]
\end{array}\right]\\
\det\left(\frac{\partial\mathbf{x}}{\partial\mathbf{y}}\right) & =\mathrm{e}^{-\sum_{i}S_{i}(\mathbf{y}_{1})}
\end{align*}
\end_inset
Note that if we use scaling layers with nonnegative output, the minus disappears.
\end_layout
\begin_layout Standard
As above, we can minimize the KL divergence, resulting in
\begin_inset Formula
\begin{align*}
J & =\log\left|\det\left(\frac{dT_{xz}}{dx}\right)\right|+\mathbb{E}_{\mathbf{z}\sim\mathcal{N}(0,I)}\left[u(T_{zx}(\mathbf{z}))\right]\\
& =-\sum_{i}S_{i}+\mathbb{E}_{\mathbf{z}\sim\mathcal{N}(0,I)}\left[u(T_{zx}(\mathbf{z}))\right]
\end{align*}
\end_inset
Where
\begin_inset Formula $i$
\end_inset
runs over
\series bold
all
\series default
scaling units in the network.
Note that in the reverse transformation